page_title: Honeycomb
section_title: Honeycomb geometry
hierarchical_section_title: Honeycomb / Honeycomb geometry
caption_attribution_description: English: Part of honeycomb where smaller worker bee cells transit into larger drone cells. As a solution, bees periodically build five-edged cells.
context_page_description: A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen.
  Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about 8.4 lb of honey to secrete 1 lb of wax, so it makes economic sense to return the wax to the hive after harvesting the honey. The structure of the comb may be left basically intact when honey is extracted from it by uncapping and spinning in a centrifugal machine—the honey extractor. If the honeycomb is too worn out, the wax can be reused in a number of ways, including making sheets of comb foundation with hexagonal pattern. Such foundation sheets allow the bees to build the comb with less effort, and the hexagonal pattern of worker-sized cell bases discourages the bees from building the larger drone cells.
  Fresh, new comb is sometimes sold and used intact as comb honey, especially if the honey is being spread on bread rather than used in cooking or as a sweetener.
  Broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet, called travel stain by beekeepers when seen on frames of comb honey.
context_section_description: The axes of honeycomb cells are always quasihorizontal, and the nonangled rows of honeycomb cells are always horizontally (not vertically) aligned.  Thus, each cell has two vertically oriented walls, with the upper and lower parts of the cells composed of two angled walls. The open end of a cell is typically referred to as the top of the cell, while the opposite end is called the bottom. The cells slope slightly upwards, between 9 and 14°, towards the open ends.
  Two possible explanations exist as to why honeycomb is composed of hexagons, rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, while minimizing the total perimeter of the cells. Known in geometry as the honeycomb conjecture, this was given by Jan Brożek and proved much later by Thomas Hales. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume.  A second reason, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency.
  The closed ends of the honeycomb cells are also an example of geometric efficiency, though three-dimensional.  The ends are trihedral (i.e., composed of three planes) sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume.  (The angle formed by the edges at the pyramidal apex, known as the tetrahedral angle, is approximately 109° 28' 16" (= arccos(−1/3)).)
  The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells.
  Individual cells do not show this geometric perfection: in a regular comb, deviations of a few percent from the "perfect" hexagonal shape occur. In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted.  Cells are also angled up about 13° from horizontal to prevent honey from dripping out.
  In 1965, László Fejes Tóth discovered the trihedral pyramidal shape (which is composed of three rhombi) used by the honeybee is not the theoretically optimal three-dimensional geometry.  A cell end composed of two hexagons and two smaller rhombuses would actually be .035% (or about one part per 2850) more efficient. This difference is too minute to measure on an actual honeycomb, and irrelevant to the hive economy in terms of efficient use of wax, considering wild comb varies considerably from any mathematical notion of "ideal" geometry.
caption: Honeycomb section containing transition from worker to drone (larger) cells – here bees make irregular and five-cornered cells (marked with red dots).
exif: {"Image ImageDescription": "", "Image Make": "NIKON", "Image Model": "E8400", "Image Orientation": "Horizontal (normal)", "Image XResolution": "72", "Image YResolution": "72", "Image ResolutionUnit": "Pixels/Inch", "Image Software": "GIMP 2.6.11", "Image DateTime": "2011:08:13 14:58:54", "Image YCbCrPositioning": "Co-sited", "Image ExifOffset": "218", "Thumbnail Compression": "JPEG (old-style)", "Thumbnail XResolution": "300", "Thumbnail YResolution": "300", "Thumbnail ResolutionUnit": "Pixels/Inch", "Thumbnail JPEGInterchangeFormat": "2134", "Thumbnail JPEGInterchangeFormatLength": "4374", "EXIF ExposureTime": "5/307", "EXIF FNumber": "29/10", "EXIF ExposureProgram": "Program Normal", "EXIF ISOSpeedRatings": "50", "EXIF ExifVersion": "0220", "EXIF DateTimeOriginal": "2011:08:05 12:39:11", "EXIF DateTimeDigitized": "2011:08:05 12:39:11", "EXIF ComponentsConfiguration": "YCbCr", "EXIF CompressedBitsPerPixel": "2", "EXIF ExposureBiasValue": "0", "EXIF MaxApertureValue": "14/5", "EXIF MeteringMode": "Pattern", "EXIF LightSource": "Unknown", "EXIF Flash": "Flash did not fire, compulsory flash mode", "EXIF FocalLength": "61/10", "EXIF FlashPixVersion": "0100", "EXIF ColorSpace": "sRGB", "EXIF ExifImageWidth": "772", "EXIF ExifImageLength": "270", "Interoperability InteroperabilityIndex": "R98", "Interoperability InteroperabilityVersion": "[48, 49, 48, 48]", "EXIF InteroperabilityOffset": "2010", "EXIF FileSource": "Digital Camera", "EXIF SceneType": "Directly Photographed", "EXIF CustomRendered": "Normal", "EXIF ExposureMode": "Auto Exposure", "EXIF WhiteBalance": "Auto", "EXIF DigitalZoomRatio": "0", "EXIF FocalLengthIn35mmFilm": "24", "EXIF SceneCaptureType": "Standard", "EXIF GainControl": "None", "EXIF Contrast": "Normal", "EXIF Saturation": "Normal", "EXIF Sharpness": "Normal", "EXIF SubjectDistanceRange": "0"}